【正文】 oy oz yysxxszzsxy?yx?xz?yz?zy?zx?plane normal direction Sense of stress ox oy oz Another specified coordinate system oxyz? ? ?Another ponent system yys??xxs??zzs??xy???yx???xz???yz???zy???zx???They can be transformed Coordinate system infinite ponent system infinite Stress tensor Determine the stress state at the point Expression of stress tensor ( matrix of tensor) －－應(yīng)力張量的表示方法（張量的矩陣形式） x x x y x zy x y y y zz x z y z zs s ss s ss s s????????x x y x zy x y y zzx zy zs ? ?? s ?? ? s????????x x x y x zy x y y y zz x z y z zs ? ?? s ?? ? s????????( , , , )ij i j x y zs ?Row Act on the same plane, but in the different direction Columm Act on the different plane, but in the same direction 1. The nine stress ponents constitute a unity which is not 2. The stress ponents depend on the choice of the coordinate 3. The stress ponents can be transformed when referring to different coordinate 相互轉(zhuǎn)換 4. The stress ponents can constitute stress invariants independent of the choice of the coordinate 可以構(gòu)成應(yīng)力不變量，該應(yīng)力不變量與坐標(biāo)系的選擇無關(guān) 5. It is a symmetric Properties of stress tensor －－應(yīng)力張量的性質(zhì) Differential equation of equilibrium in the neighbourhood of a point x z y x?z?y?A B C D E F G O Force equilibrium (力平衡 ) xsxy?xz?ysyx?yz?zszx?zy?Take an element using section method. Analyses the stresses on the face of the element. Stress ponents are the continuous function of the Cartesian coordinate. ? ?0 ,ij ij x y zss?（力平衡微分方程） ? ?,Aij ij x x y y z zs s ? ? ?? ? ? ?AijsExpand into Taylor’s series: ? ? ? ?, , , , i j i j i ji j i jx x y y z z x y z x y zx y zs s ss ? ? ? s ? ? ?? ? ?? ? ? ? ? ? ? ?? ? ?Neglect the terms in higher powers of x? y? z?and ? ? ? ?, , , , i j i j i ji j i jx x y y z z x y z x y zx y zs s ss ? ? ? s ? ? ?? ? ?? ? ? ? ? ? ?? ? ?At the point A (O A). ,x y z? ? ?incremet : ? ? ? ?, , , , i j i jBi j i j i jx x y y z x y z x yxysss s ? ? s ? ???? ? ? ? ? ???At the point B (O B). ,xy??incremet : x z y x?z?y?A B C D E F G O At the point G (O G). x?incremet : ? ? ? ?, , , , ijGi j i j i jx x y z x y z xxss s ? s ??? ? ? ? ?x z y x?z?y?A B C D E F G O xsxy?xz?ysyx?yz?zszx?zy?xx xxss??? ?xyxy xx????? ?xzxz xx????? ?yy yyss??? ?yxyx yy????? ?yzyz yy????? ?zz zzss??? ?zxzx zz????? ? zyzy zz????? ?x z y x?z?y?A B C D E F G O 0X ??Consider that the stress on each face are uniform (infinitesimal) 000XYZ? ?? ??? ?????The element is in equilibrium xxxx x y zxss ? ? ??????????zxzx z x yz?? ? ? ???????????0yxx x z xx y z?s????? ? ?? ? ?x z y x?z?y?A B C D E F G O xsxy?xz?ysyx?yz?zszx?zy?xx xxss??? ?xyxy xx????? ?xzxz xx????? ?yy yyss??? ?yxyx yy????? ?yzyz yy????? ?zz zzss??? ?zxzx zz????? ? zyzy zz????? ?xx yzs ? ?? yxyx y x zy?? ? ? ??????????? yxxz? ? ??yx xy? ? ??0?0yxx x z xx y z?s????? ? ?? ? ?0x y y y z yx y z? s ?? ? ?? ? ?? ? ?0yzxz zzx y z?? s?? ?? ? ?? ? ?0ijixs? ??By means of tensor notation: j : is free subscript, it appears only one time in one term and is the same in all terms, it is replaced by x, y and z cyclicly in different eqs. i : is dummy subscript, it appears twice in one term and it is considered as the sum of three terms, in which the sub is replaced by x, y and z cyclicly . 0ijixs? ??0x j y j z jx y zx x xs s s? ? ?? ? ?? ? ?,jx? 0yxx x z xx y zsss???? ? ?? ? ?,jy? 0x y y y z yx y zs s s? ? ?? ? ?? ? ?,jz? 0yzxz zzx y zss s?? ?? ? ?? ? ?ii x x y y zzs s s s? ? ?i ij jSls?dummy , , 。 The problem is how to determine the principal stress (magnitude and directions) 問題是如何確定主應(yīng)力（大小和方向） Suppose that the direction cosines of the normal to the principal plane are l, m, n. and Sn = SR on the principal plane. Resultant vector ponent vector j n j n ij iS S l S l???lSS nx ?nSmSS ny ?nSS nz ?stress on oblique plane stress on three Cartesian plane: nmlS zxyxxxx ??s ???nmlS zyyyxyy ?s? ???nmlS zzyzxzz s?? ??? ? ? 0?? jnijij lS?si = j i = j ???? 01ij? kronecker delta ? ? 0???? nmlS zxyxnxx ??s? ? 0???? nmSl zynyyxy ?s?? ? 0???? nSml nzzyzxz s??For the equation to have a nontrivial solution for l, m, n. the determinant of the coefficients must vanish: 0x x n y x z xij ij n x y y y n z yx z y x z z nSSSSs ? ?s ? ? s ?? ? s?? ? ? ??After expanding the determinant and rearranging the terms the coefficients matrix is as follows: . ? ? ? ?3 2 2 2 2n n x x y y z z n x x y y y y z z z z x x x y y z z xS S Ss s s s s s s s s ? ? ???? ? ? ? ? ? ? ? ???? ?22220x x y y z z x y y z z x y z x x z x y y x y z zs s s ? ? ? ? s ? s ? s? ? ? ? ? ?Trivial solution（無效解） is l = m = n = 0, violate（違背） the relation 2 2 2 1l m n? ? ?為了使得以 l、 m、 n 為變量的方程有非零解，其系數(shù)矩陣必須為零。 l3, m3, n3。 這三組方向余弦便確定了三個(gè)相互垂直的主方向 . Stress invariants （應(yīng)力不變量） At a point of a body (given stress state) ijs 321 J,J,J 032213 ???? JSJSJS nnn321 sss ,ji ??s 321 ??? J,J,J 032213 ???? ??? JSJSJS nnnTherefore 11 JJ ?? 22 JJ ?? 33 JJ ??zzyyxxzzyyxxJ ?????? ?????? ssssss1 First (linear ) invariant （第一不變量） ? ? ? ?2222 zxyzxyxxzzzzyyyyxxJ ???ssssss ???????? ? ? ?222 xzzyyxxxzzzzyyyyxx ?????????????????? ??????? ???ssssssSecond (quadratic) invariant For a given stress state s1, s2, s3 are unique, independent of the choice of coordinate system ? ?zz2xyyy2zxxx2yzzxyzxyzzyyxx3 2J s?s?s????sss ?????? ?zzyxyyxzxxzyxzzyyxzzyyxx ??????????????????????? ????? s?s?s??